Sturridge left Liverpool after six years last summer but has made just 11 Super Lig appearances for his new club this season, scoring four goals.
According to the Birmingham Mail, the 30-year-old is set to depart the Turkish club in the summer and Dean Smith's side are especially interested.
The former Chelsea and Manchester City forward was linked with Villa last summer and is said to still be on the club's radar.
Since arriving at Trabzonspor his game time has been limited due to injury, and he has made just three appearances in all competitions since December.
Villa have also suffered injuries in their forward line, with Wesley ruled out for the rest of the season.